If Sgt. Doakes really thought Dexter was a psycho and up to something he wouldn't have blatantly followed him, he would have done it secretly, with a camera, and Dexter would have been caught from the start.
For a serial killer who has been trained from childhood to be meticulous, Dexter repeatedly makes amateurish and ridiculously stupid mistakes, not to mention revealing who he really is and who he has killed to numerous people.
In every crime scene throughout the entire series both Forensics and the detectives handle the body before the medical examiner arrives. In every state in the country, no one can touch the body until the medical examiner arrives.
Throughout the series the forensics team are seen handling crime scene evidence, walking through evidence, and/or climbing through evidence without wearing proper protective gear so they don't contaminate the crime scene.
In the main title, there are lots of "continuity errors", where you can see the same things being done over and over again. This is clearly intentional from the producers, as it shows that Dexter is following a strict routine, doing the same things every morning.
In Season 2 Rita absolutely insists that Dexter work a recovery program and remain clean and sober as a condition to them remaining together. In Seasons 3 and 4 this does not come up at all plus they are in multiple drinking situations and Rita says nothing about Dexter drinking.